The fundamental theorem of contour integration says if one has a function and its antiderivative, and integrates the function over a closed loop the result is zero.
Cauchy's theorem (Goursat's Version) says the integral of a function in a holomorphic domain in a closed loop is zero.
Cauchy's theorem is apparently much stronger, the proof is certainly more intricate. Can someone please give trivial and nontrivial examples of integrals that Cauchy's Theorem applies to that FTCI does not?
Having proven Cauchy's Theorem, is the FTCI useful for anything, anymore?
Best Answer
I would state the "fundamental theorem of contour integration" as follows.
Fundamental Theorem of Contour Integration: Let $U\subseteq\mathbb{C}$ be an open set, and let $f\colon U\to \mathbb{C}$ be a continuous function. Then the following conditions are equivalent:
Notice that the theorem does not say that all holomorphic functions necessarily satisfy the two equivalent conditions. In fact, there are situations where they won't. For instance, if $U = \mathbb{C}\smallsetminus\{0\}$ and $f(z) = 1/z$, then $f$ does not have an antiderivative on $U$, even though it is holomorphic. The Cauchy-Goursat theorem tells us one situation in which holomorphic functions are guaranteed to satisfy conditions (1) and (2).
Cauchy-Goursat: Let $U\subseteq\mathbb{C}$ be a simply connected open set. Then all holomorphic functions $f\colon U\to \mathbb{C}$ satisfy the equivalent conditions of the previous theorem.